Like everyone else has said, those tweeters are likely the issue. I know exactly what you are describing and I didn't buy them for that very reason when I was demoing them.
There are a lot of good options out there that will work very well with that Moon amp and the balance of your system. Key is doing some listening.
I personally would look at Dynaudio or Canton as they are big companies who deliver a strong value and have a smoother sound. A used pair of Special 40s or Reference 9Ks would be insanely good.
Full disclosure, I have no relationship with Dynaudio but am a Canton dealer.